In general, all phenomenologically viable lepton mass matrices with
with parallel texture/cofactor zeros can be realized in seesaw
models with Abelian flavor symmetry. The lepton mass matrices of
class I are equivalent to the ones with no-parallel texture zeros.
The symmetry realization of such texture structures has been
performed in Ref.[1]}. Thus we only consider class II and III.
In this section, we take the first matrix pattern of class II as a
illustration. It is shown that the lepton mass matrix can be
realized based on the type-I seesaw models with the \(Z_{4}\times Z_{2}\)  flavor symmetry. We take the same strategy of
Ref.[2]}, [3]}, [4]}. In flavor basis, \(M_{\nu }\)  belonging
to class II is realized under \(Z_{8}\)  symmetry[3]}.
Different from Ref.[3]}, we build the model under the basis
where \(M_{l}\)  is nodiagonal. Under the \(Z_{4}\times Z_{2}\)  symmetry,
the three charged lepton doublets \(D_{i L}=(\nu _{i L}, l_{iL})\) ,
three right-handed charged lepton singlets \(l_{iR}\)  and three
right-handed neutrinos \(\nu _{iR}\)  (where \(i=e,\mu ,\tau \) ) transform
as
\(\begin{split}\nu _{eR}\sim (\omega ,1),\quad \quad \nu _{\mu R}\sim (1,1), \quad \quad \nu _{\tau R}\sim (\omega ^{2},1)\\D_{eL}\sim (\omega ,-1),\quad \quad D_{\mu L}\sim (1,-1),\quad \quad D_{\tau L}\sim (\omega ^{2},-1)\\l_{eR}\sim (\omega ^{3},-1), \quad \quad l_{\mu R}\sim (1,-1),\quad \quad l_{\tau R}\sim (\omega ^{2},-1)\end{split}\) 
